Question

Graphing sine and cosine functions Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. $$g(x)=-2 \cos (x / 3)$$

Short answer

Question: Determine the amplitude, period, phase shift, and vertical shift for the function \(g(x) = -2 \cos(x/3)\) and briefly describe the process of sketching its graph. Answer: For the given function \(g(x) = -2 \cos(x/3)\), the amplitude is 2, the period is \(6\pi\), the phase shift is 0, and the vertical shift is 0. To sketch the graph, start with the basic cosine function, scale vertically by a factor of 2, compress horizontally by a factor of 3, and then reflect the graph across the x-axis.

Step-by-step solution

Identify the Amplitude

The formula for this cosine function is given by \(g(x) = A\cos(Bx + C) + D\). In our case, \(g(x) = -2\cos(x/3)\), which means the amplitude A is -2. However, amplitude is always a positive value, so the amplitude of the given function is 2.

Determine the Period

Recalling the formula for the period of a cosine function is \(P=\frac{2\pi}{|B|}\). In our given function, \(B=\frac{1}{3}\). Thus, the period of g(x) is: \(P = \frac{2\pi}{|\frac{1}{3}|} = \frac{2\pi}{\frac{1}{3}} = 2\pi \cdot 3 = 6\pi\).

Identify the Phase Shift

The phase shift can be calculated using \(C\) from the general formula \(g(x) = A\cos(Bx + C) + D\). In our case, there is no addition or subtraction inside the parentheses, which means the phase shift is 0.

Determine the Vertical Shift

The vertical shift can be found using the value of \(D\) in the general formula. In our particular case, there is no addition or subtraction outside the cosine function, which implies the vertical shift is 0.

Sketch the Graph

Start with the basic cosine function, \(y=\cos x\). Then apply the following transformations sequentially: 1. Scale vertically by a factor of 2 (amplitude). 2. Compress horizontally by a factor of 3 (period). Since there are no phase and vertical shifts, we don't need to apply any translations to the graph. Finally, reflect the graph across the x-axis because the amplitude is negative, which gives us the graph of \(g(x)=-2 \cos (x / 3)\).

Check with a Graphing Utility

To check our work, we will use a graphing calculator or an online tool such as Desmos. Input the given function \(g(x) = -2 \cos(x/3)\), and compare it with the graph we sketched. The shape and placement of the graph should be consistent with our manually created graph. After confirming that both graphs are similar, we can be confident in our solution.

Key concepts

Amplitude of Trigonometric Functions

When we talk about the amplitude of trigonometric functions, we are referring to the height of the wave from the middle of the wave to its peak or trough (in other words, how 'tall' or 'short' the wave is). The amplitude is an absolute value, meaning that it’s always taken as a positive number, even if the trigonometric function is preceded by a negative coefficient.

For example, in the function \(g(x) = -2\cos(x/3)\), the coefficient -2 affects the amplitude of the cosine wave. Although the amplitude is represented by the number 2 (the absolute value of the coefficient), the negative sign indicates that the graph will reflect across the x-axis. Visualizing this is key to understanding the wave's shape. The amplitude is crucial for graphing because it tells you the vertical stretch of the wave.

Period of Cosine Function

The period of a trigonometric function, like the cosine function, is the length of one complete cycle on the graph before the pattern repeats. For \(y = \cos x\), the basic period is \(2\pi\), because cosine waves repeat every \(2\pi\) radians. However, this period changes when the function is scaled horizontally.

In our example, the function \(g(x) = -2\cos(x/3)\) has a period of \(6\pi\). This is because the horizontal scaling factor, \(B = 1/3\), stretches the graph so that one full wave is longer, specifically three times the length of the basic cosine wave's period. Hence, the importance of the period is that it informs us about the horizontal stretch or compression of the wave.

Phase Shift in Trigonometry

Phase shift in trigonometry describes the horizontal shifting of a graph left or right. This can completely change where the wave starts on the x-axis. Phase shifts are identified by the value \(C\) in the expressions of the form \(A\cos(Bx+C)+D\) or \(A\sin(Bx+C)+D\), where the graph is shifted to the left by \(C/B\) units if \(C\) is positive and to the right by \(C/B\) units if \(C\) is negative.

In our function \(g(x)=-2\cos(x/3)\), there is no value added or subtracted inside the cosine function, which indicates the phase shift is 0. The graph will start at the usual starting point for a cosine wave, which is at a peak when \(x=0\). Understanding the phase shift is potentially transformational for graphing sine and cosine functions as it dictates the starting point of the wave along the x-axis.

Transformations of Trigonometric Graphs

Transformations of trigonometric graphs involve shifting, stretching, or reflecting the basic sine and cosine graphs. The transformations are determined by the coefficients and constants in the trigonometric equation. There are four main types of transformations:

• Vertical stretch/compression is governed by the amplitude.
• Horizontal stretch/compression is determined by the period.
• Horizontal shifting is controlled by the phase shift.
• Vertical shifting is due to up or down movements and is represented in the equation by \(+D\) or \(-D\).

In our example with \(g(x) = -2\cos(x/3)\), we can systematically apply these transformations to the base cosine graph. There is a vertical stretch of factor 2 and horizontal stretch by factor 3. Since the amplitude has a negative value, we also apply a reflection across the x-axis. There is no horizontal or vertical shifting to consider. The order in which we apply these transformations can impact the accuracy of the resulting graph, so this methodical approach is critical for students to grasp.